If you start using Claw on Foof in Round 19 instead of passing in between cooldowns, you can end the fight several rounds earlier—your BONESTORM in Round 25 should kill all three dogs at the same time.

(Note: if you crit Claw more than once on Foof, there's a chance he may die early. This is fine. The Round 25 BONESTORM will still kill Lil' Sizzler on the backline even if Stumpers dodges it, and you're still at full health—you'd have to miss more than 10 times in a row for it to be an issue.)

This strategy is slow, but it did the job for me. Like Eekwibble, I try to level at least one pet at least a little bit in every battle that I can. There may be a way to do that against this team of (almost) indomitable Corgis, but I don't see it at this point. Certainly not with an entire team of Undead. Both of my back-line pets expired and my Hatchling was just a couple of HP away from his rez round. Tough, tough puppies. Thanks for the strategy.

There are a few mentions of Blightbreath here but none that say an H/P breed can solo the fight... in 14 turns...
I'm a little annoyed, tbh, because I like looking for ways to level pets on fights like these that give decent xp, but, Blightbreath (1-2-1) was so effective, I didn't get a chance to rotate a levelling pet in.
Anyway, here's the rotation;

I received no crits and dealt only 2, they were on Stumpers on turn 3 and on Lil' Sizzle on turn 9. I did the calculations and without those Dreadful Breath crits, Stumpers would still have died (receiving 1403 damage to his 1400hp) and Lil' Sizzle would have been alive with 38hp. Blightbreath was still alive with its Undead racial unused, so it's clearly soloable by an H/P breed Blightbreath alone, even without crits.
I'll try and remember to check out a levelling pet strat the next time she's up.

If you are like me and don't have the hatchling.
What Evil suggested, Mr Bigglesworth 1,1,1 Fragment of Desire 1,1,1 and Blighthawk 1,1,1.
Worked well for me.
Ice tomb, claw as needed. Ice tomb in undead round.
Spiritfire to finish 2nd pet or could use soulrush might make it faster.
Reflective shield on Little Sizzle. Then spritfire bolt. Don't waste your soulrush here like I did.
Infected claw and liftoff with Blighthawk after switch finished him off.
Thanks Evil for the strat :-)

It might be worth mentioning that floof's dreadful breath spreads it's damage across your team so if your backline pets die quickly your fossilized hatchling will take a lot more damage. After I realized this I chose my two highest hp pets for my backline and had no problems. The strat didn't work for me on multiple attempts before I made this change.

funny-i died first try so i also put Boneshard in at turn 3, bonestorm times 3. He died and FH came back to continue his rotation. Then when back corgis almost dead i started claw so Foof was only 2 hits away from dead. I had high health scourge whelpling on standby but I didn't need.

I used Blightbreath, Ghastly Kid, and Scourged Whelpling. Acid Rain and Dreadful Breath two cycles will knock out the fire puppy. and lower the others. when Blight dies come in with very low health Whelpling finish off the other dog Foof . and then finally end up with ghastly kid evading Stumpers attack and then killing Stumpers with Diseased Bite. for FFH

Both of my backline pets died, and on turn 35 the hatchling and Foof killed each other, didn't get credit. Complete waste of time as written. Choose two high HP backline pets, I picked Unborn Val'kyr and Grotesque. Every other turn replace one of the passes for claw so long as Foof isn't under 300 health.